TITLE: Solving for unknown
QUESTION [7 upvotes]: When solving for unknowns why does the symbols sometimes change from $a + $ to $a -$ or vice versa or stays the same?
For example (using solutions from text): 
$2x+6 = x+16\quad$ and the solution is $2x-x = 16-6\quad \Rightarrow \quad x=10$.
Or
$8x-12 = -3x+21\quad $Solution: $8x+3x = 21+12\quad \Rightarrow \quad
                 11x = 33\quad \Rightarrow \quad
                 x=3$

REPLY [2 votes]: First, you must understand that if we have a statement like $n = m$, then $n$ equals $m$. That means that $n + 1 $ is STRICTLY equal to $m$. That also means that $2m = 2n$, ${n \over 2} = {m \over 2}$ and whatnot. 
What you always do in linear equations is, in fancy term, “isolate” the variable. What is isolation? Isolating a variable is to perform certain actions to both sides (important) in order to have a variable on one side and a constant on another, if possible. I have an example later, but before... 
Your basic rule is that you have to do the same thing on both sides. You can do anything, but for it to lead you to an answer, you have to follow a suggested rule: “$\rm AS-MD$”

Addition and subtraction in any order.
Division and multiplication in any order.

Note: you apply this rule to the variable side—the side which has your variable required to be solved for.
Let's have an example now. $$2x  + 1 = 5$$An effective way is to subtract one from both sides according to the rule.$$\begin{align}2x + 1 - 1& = 5 - 1 \\ 2x&=4\end{align}$$Now that we have nothing to subtract or add, let's try the multiply/divide part. Yes, we can divide $2$ from both sides. $${2x \over 2} = {4 \over 2} \quad \Rightarrow \quad x = 2$$

Now, that you get the basic idea, let's try to solve the same equation through the way you suggested. $$\begin{align}2x + 1 &= 5 \\ \\ 2x & = 5 - 1 \\ \\ 2x & = 4 \\ \\ x &= {4 \over 2} \\ \\ x & = 2 \end{align}$$ Now, I cannot prove rigorously, but I can tell you that the sign changing seems something fancy but is based on the idea of doing the same things on both sides. The intution: $$a + m = b \quad \Rightarrow \quad a = b-m  $$ Which is derived by, $$\begin{align} a + m - m & = b - m \\ \\ a + \underbrace{( m - m)}_{\text{always zero}}& = b - m \\ \\ a &= b - m\end{align}$$